Linear methods for non-linear inverse problems

TL;DR

This paper introduces a Bayesian approach to solve non-linear inverse problems involving PDEs by transforming them into linear inverse problems, achieving optimal recovery rates and adaptive estimation without prior smoothness knowledge.

Contribution

It develops a Bayesian framework for non-linear PDE inverse problems, providing theoretical guarantees and adaptive priors for optimal recovery of the unknown function.

Findings

Bayesian methods achieve optimal recovery rates for both the solution and the unknown function.

Adaptive priors enable rate adaptation without smoothness assumptions.

Numerical experiments confirm theoretical results.

Abstract

We consider the recovery of an unknown function $f$ from a noisy observation of the solution $u_f$ to a partial differential equation that can be written in the form $\mathcal{L} u_f=c(f,u_f)$, for a differential operator $\mathcal{L}$ that is rich enough to recover $f$ from $\mathcal{L} u_f$. Examples include the time-independent Schr\"odinger equation $\Delta u_f = 2u_ff$, the heat equation with absorption term $(\partial_t -\Delta_x/2) u_f=fu_f$, and the Darcy problem $\nabla\cdot (f \nabla u_f) = h$. We transform this problem into the linear inverse problem of recovering $\mathcal{L} u_f$ under the Dirichlet boundary condition, and show that Bayesian methods with priors placed either on $u_f$ or $\mathcal{L} u_f$ for this problem yield optimal recovery rates not only for $u_f$, but also for $f$. We also derive frequentist coverage guarantees for the corresponding Bayesian credible sets. Adaptive priors are shown to yield adaptive contraction rates for $f$, thus eliminating the need to know the smoothness of this function. The results are illustrated by numerical experiments on synthetic data sets.